Can I take my daughter and move out of state to be closer to my family?

I practice in Texas and am not licensed to practice in Ohio. Ohio law would control regarding the rights and duties of the parents. That being said, in many jurisdictions the following would apply:

If there is a current court order establishing the father's right those orders would be controlling. Paternity orders often include specific possession rights for the father and not infrequently a geographical restriction.

If there are no orders, or if there are orders with no geographic restriction, you can leave the state and his permission would not be required.

Under Ohio law, unmarried mothers are the sole custodians of their children. Even if paterntiy was estabished by affidavit at time of birth, this does not give the fathers any parental rights over children. To do so, the father must obtain a court order outlining his rights to parenting time with a child. If there is no court order, then you are able to move with the child without the consent of the father. If there is an order, you need to file a motion to modify the order to allow you to move with the child to a new jurisdiction.